3.29.79 \(\int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} (-a^4+4 a^3 x+(-6 a^2+b^2 d) x^2+2 (2 a-b d) x^3+(-1+d) x^4)} \, dx\) [2879]

3.29.79.1 Optimal result

Integrand size = 83, antiderivative size = 310 \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{2 a-2 x+\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{-2 a+2 x+\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{a-x}\right )}{d^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}}{a^2-2 a x+x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{5/6}} \]

output

-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(2*a-2*x+ 
d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)))/d^(5/6)+1/2*3^(1/2)*arctan(3^(1/2)* 
d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(-2*a+2*x+d^(1/6)*(a*b*x+(-a-b)*x^2+x 
^3)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(a-x))/d^ 
(5/6)+1/2*arctanh((a*d^(1/6)-d^(1/6)*x)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(a^2- 
2*a*x+x^2+d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)))/d^(5/6)
 

3.29.79.2 Mathematica [F]

\[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx \]

input

Integrate[(x*(-b + x)*(a*b + (-2*a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*( 
-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)) 
,x]
 

output

Integrate[(x*(-b + x)*(a*b + (-2*a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*( 
-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)) 
, x]
 

3.29.79.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x (x-b) (x (b-2 a)+a b)}{\sqrt [3]{x (x-a) (x-b)} \left (-a^4+4 a^3 x+x^2 \left (b^2 d-6 a^2\right )+2 x^3 (2 a-b d)+(d-1) x^4\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {(b-x) x^{2/3} (a b-(2 a-b) x)}{\sqrt [3]{x^2-(a+b) x+a b} \left (a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2\right )}dx}{\sqrt [3]{x (a-x) (b-x)}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {(b-x) x^{4/3} (a b-(2 a-b) x)}{\sqrt [3]{x^2-(a+b) x+a b} \left (a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\)

\(\Big \downarrow \) 1395

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(b-x)^{2/3} x^{4/3} (a b-(2 a-b) x)}{\sqrt [3]{a-x} \left (a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-2 a) (b-x)^{2/3} x^{7/3}}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4\right )}+\frac {a b (b-x)^{2/3} x^{4/3}}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} \sqrt [3]{b-x} \left (a b \int \frac {(b-x)^{2/3} x^{4/3}}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4\right )}d\sqrt [3]{x}-(2 a-b) \int \frac {(b-x)^{2/3} x^{7/3}}{\sqrt [3]{a-x} \left (a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{x (a-x) (b-x)}}\)

input

Int[(x*(-b + x)*(a*b + (-2*a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(-a^4 + 
 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)),x]
 

output

$Aborted
 

3.29.79.3.1 Defintions of rubi rules used

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.29.79.4 Maple [F]

\[\int \frac {x \left (-b +x \right ) \left (a b +\left (-2 a +b \right ) x \right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{4}+4 a^{3} x +\left (b^{2} d -6 a^{2}\right ) x^{2}+2 \left (-b d +2 a \right ) x^{3}+\left (-1+d \right ) x^{4}\right )}d x\]

input

int(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+(b^2*d 
-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x)
 

output

int(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+(b^2*d 
-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x)
 

3.29.79.5 Fricas [F(-1)]

Timed out. \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]

input

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+ 
(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x, algorithm="fricas")
 

output

Timed out
 

3.29.79.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]

input

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))**(1/3)/(-a**4+4*a**3 
*x+(b**2*d-6*a**2)*x**2+2*(-b*d+2*a)*x**3+(-1+d)*x**4),x)
 

output

Timed out
 

3.29.79.7 Maxima [F]

\[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a b - {\left (2 \, a - b\right )} x\right )} {\left (b - x\right )} x}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 2 \, {\left (b d - 2 \, a\right )} x^{3} + {\left (b^{2} d - 6 \, a^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+ 
(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x, algorithm="maxima")
 

output

-integrate((a*b - (2*a - b)*x)*(b - x)*x/(((d - 1)*x^4 - a^4 + 4*a^3*x - 2 
*(b*d - 2*a)*x^3 + (b^2*d - 6*a^2)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)
 

3.29.79.8 Giac [F]

\[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a b - {\left (2 \, a - b\right )} x\right )} {\left (b - x\right )} x}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 2 \, {\left (b d - 2 \, a\right )} x^{3} + {\left (b^{2} d - 6 \, a^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+ 
(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x, algorithm="giac")
 

output

integrate(-(a*b - (2*a - b)*x)*(b - x)*x/(((d - 1)*x^4 - a^4 + 4*a^3*x - 2 
*(b*d - 2*a)*x^3 + (b^2*d - 6*a^2)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)
 

3.29.79.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int -\frac {x\,\left (a\,b-x\,\left (2\,a-b\right )\right )\,\left (b-x\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^2\,\left (b^2\,d-6\,a^2\right )+2\,x^3\,\left (2\,a-b\,d\right )+4\,a^3\,x-a^4+x^4\,\left (d-1\right )\right )} \,d x \]

input

int(-(x*(a*b - x*(2*a - b))*(b - x))/((x*(a - x)*(b - x))^(1/3)*(x^2*(b^2* 
d - 6*a^2) + 2*x^3*(2*a - b*d) + 4*a^3*x - a^4 + x^4*(d - 1))),x)
 

output

int(-(x*(a*b - x*(2*a - b))*(b - x))/((x*(a - x)*(b - x))^(1/3)*(x^2*(b^2* 
d - 6*a^2) + 2*x^3*(2*a - b*d) + 4*a^3*x - a^4 + x^4*(d - 1))), x)